polyomino

There is one monomino, a square.

There is one domino, a 1x2 rectangle.

There are two distinct triominoes, a straight chain of three squares, and a bent one. The bent triomino is the basis for Sid Sackson's game Tromino Go.

There are five distinct tetrominoes, or seven if you count reflections as distinct. These seven pieces were the basis for the game Tetris.

There are twelve distinct pentominoes. These pieces are the subject of many games and puzzles (especially tiling puzzles) in recreational mathematics.

For concreteness, we'll consider polyominoes distinct if they're (nonsymmetrical) reflections; this can only throw our count off by a factor of 2, of course. These are also called "animals".

There are exponentially many n-square polyominoes. Indeed, if we pick a square and proceed (for n-1 steps) by picking either the next square up or the next square right, we'll get 2^n-1 polyominoes that are distinct, except maybe they can be rotated onto each other. So there are at least 2^n-3 polyominoes -- exponentially many.

On the other hand, note that every self-avoiding walk on the lattice has a polyomino as its image, so there are no more than 4×3^n-1 polyominoes.

We'd like to say there are Θ(bⁿ) polyominoes; what is b?